A121173 Sequence S with property that for n in S, a(n) = a(1) + a(2) +...+ a(n-1) and for n not in S, a(n) = n+1.

2, 2, 4, 8, 6, 22, 8, 52, 10, 114, 12, 240, 14, 494, 16, 1004, 18, 2026, 20, 4072, 22, 8166, 24, 16356, 26, 32738, 28, 65504, 30, 131038, 32, 262108, 34, 524250, 36, 1048536, 38, 2097110, 40, 4194260, 42, 8388562, 44, 16777168, 46, 33554382

Offset: 1

Author

Max Alekseyev, Aug 15 2006

Keywords

• nonn

Comments

a(1)=1 cannot happen, so the sequence S starts with a(1)=2.

Note that a(n)=a(1)+a(2)+...+a(n-1) can hold even if n is not in S. The smallest example is n=3.

All terms are even. - Reinhard Zumkeller, Nov 06 2013

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A105753, A121053, A121174, A121175, A145654.

Programs

Haskell

a121173 n = a121173_list !! (n-1)
a121173_list = f 1 [] where
   f x ys = y : f (x + 1) (y : ys) where
     y = if x `elem` ys then sum ys else x + 1
-- Reinhard Zumkeller, Nov 06 2013

Mathematica

s={2};Do[If[MemberQ[s,n],m=Total[s],m=n+1];AppendTo[s,m],{n,2,46}];s (* James C. McMahon, Oct 13 2024 *)

Python

from itertools import count, islice
def agen(): # generator of terms
    S, s, an = {2}, 2, 2
    for n in count(2):
        yield an
        an = s if n in S else n+1
        s += an
        S.add(an)
print(list(islice(agen(), 50))) # Michael S. Branicky, Oct 13 2024

Formula

a(2*n) = A145654(n+1). - Reinhard Zumkeller, Nov 06 2013

a(2*n+1) = 2*n+2.

From Colin Barker, Jan 30 2016: (Start)

a(n) = 2*(2^(n/2+1)-2)-n for n even.

a(n) = n+1 for n odd.

a(n) = -a(n-1)+3*a(n-2)+3*a(n-3)-2*a(n-4)-2*a(n-5) for n>5.

G.f.: 2*x*(1+2*x) / ((1-x)*(1+x)^2*(1-2*x^2)). (End)

E.g.f.: (x - 4)*cosh(x) + 4*cosh(sqrt(2)*x) + (1 - x)*sinh(x). - Stefano Spezia, Oct 14 2024